A liquid of density `rho` is in a vessel rotating with angular velocity `omega` as shown in figure. If `P_0` is the atmospheric pressure, find pressure at a radial distance `r` from the axis.
.

Explanation

Consider a fluid particle `A` of mass at position `(x,y)`
Forces acting on particle `A` are
(i) pseudo force `(m omega^2 x)`
(ii) weight `(mg)`
Net force on `A` should be perpendicular to the free surface (in equilibrium) [liquid cannot take shear stress]
`tan theta = (m omega^2 x)/(mg) = (omega^2 x)/(g)`
`(dy)/(dx) = (omega^2 x)/(g)`
`int_0^y dy = (omega^2)/(g) int_0^x x dx`
`y = (omega^2 x^2)/(2 g)`
At `x = r, y=(omega^2 r^2)/(g)`
`P = P_0 + rho gy = P_0 + (rho omega^2 r^2)/(2) = P_0 + (1)/(2) rho v^2 ("since" v = r omega)`.